Advertisements
Advertisements
प्रश्न
\[\int \sin^3 x \cos^4 x\ \text{ dx }\]
योग
Advertisements
उत्तर
\[\text{ Let I }= \int \sin^3 x \cdot \cos^4 x\ dx\]
\[ = \int \sin^2 x \cdot \sin x \cdot \cos^4 x\ dx\]
\[ = \int\left( 1 - \cos^2 x \right) \cdot \cos^4 x \cdot \sin x\ dx \]
\[ = \int\left( \cos^4 x - \cos^6 x \right) \cdot \sin x\ dx\]
\[\text{ Putting cos x = t}\]
\[ \Rightarrow - \sin x\ dx = dt\]
\[ \Rightarrow \sin x\ dx = - dt\]
\[ \therefore I = - \int\left( t^4 - t^6 \right)dt\]
\[ = \int\left( t^6 - t^4 \right)dt\]
\[ = \frac{t^7}{7} - \frac{t^5}{5} + C\]
\[ = \frac{\cos^7 x}{7} - \frac{\cos^5 x}{5} + C......... \left[ \because t = \cos x \right]\]
shaalaa.com
क्या इस प्रश्न या उत्तर में कोई त्रुटि है?
APPEARS IN
संबंधित प्रश्न
\[\int\frac{x^6 + 1}{x^2 + 1} dx\]
\[\int \left( \tan x + \cot x \right)^2 dx\]
\[\int\frac{\cos x}{1 - \cos x} \text{dx }or \int\frac{\cot x}{\text{cosec } {x }- \cot x} dx\]
\[\int \tan^{- 1} \left( \frac{\sin 2x}{1 + \cos 2x} \right) dx\]
\[\int\frac{1}{\sqrt{x + 3} - \sqrt{x + 2}} dx\]
\[\int\frac{2x - 1}{\left( x - 1 \right)^2} dx\]
` ∫ sin 4x cos 7x dx `
\[\int\sqrt{\frac{1 - \sin 2x}{1 + \sin 2x}} dx\]
\[\int\frac{e^x + 1}{e^x + x} dx\]
\[\int\frac{\sin 2x}{\sin \left( x - \frac{\pi}{6} \right) \sin \left( x + \frac{\pi}{6} \right)} dx\]
\[\int \sin^5\text{ x }\text{cos x dx}\]
\[\int\frac{2x - 1}{\left( x - 1 \right)^2} dx\]
\[\int \cot^6 x \text{ dx }\]
\[\int \sin^5 x \text{ dx }\]
\[\int\frac{1}{\sqrt{\left( 2 - x \right)^2 + 1}} dx\]
\[\int\frac{x}{3 x^4 - 18 x^2 + 11} dx\]
\[\int\frac{1}{\sqrt{\left( x - \alpha \right)\left( \beta - x \right)}} dx, \left( \beta > \alpha \right)\]
\[\int\frac{\cos 2x}{\sqrt{\sin^2 2x + 8}} dx\]
\[\int\frac{x + 1}{x^2 + x + 3} dx\]
\[\int\frac{1}{\sqrt{3} \sin x + \cos x} dx\]
\[\int\frac{1}{p + q \tan x} \text{ dx }\]
` ∫ sin x log (\text{ cos x ) } dx `
\[\int \left( \log x \right)^2 \cdot x\ dx\]
\[\int \cos^{- 1} \left( 4 x^3 - 3x \right) \text{ dx }\]
\[\int \tan^{- 1} \left( \sqrt{x} \right) \text{dx }\]
\[\int e^x \left[ \sec x + \log \left( \sec x + \tan x \right) \right] dx\]
\[\int e^x \frac{\left( 1 - x \right)^2}{\left( 1 + x^2 \right)^2} \text{ dx }\]
\[\int e^x \cdot \frac{\sqrt{1 - x^2} \sin^{- 1} x + 1}{\sqrt{1 - x^2}} \text{ dx }\]
\[\int\left( x + 2 \right) \sqrt{x^2 + x + 1} \text{ dx }\]
\[\int\frac{x^2 + 1}{x^2 - 1} dx\]
\[\int e^x \left( 1 - \cot x + \cot^2 x \right) dx =\]
\[\int\frac{1}{7 + 5 \cos x} dx =\]
\[\int \cot^5 x\ dx\]
\[\int\frac{\sqrt{a} - \sqrt{x}}{1 - \sqrt{ax}}\text{ dx }\]
\[\int\frac{6x + 5}{\sqrt{6 + x - 2 x^2}} \text{ dx}\]
\[\int\frac{1 + x^2}{\sqrt{1 - x^2}} \text{ dx }\]
\[\int x^2 \tan^{- 1} x\ dx\]
\[\int\frac{x^2}{x^2 + 7x + 10} dx\]
\[\int\frac{3x + 1}{\sqrt{5 - 2x - x^2}} \text{ dx }\]
